BRMS models’ validation
This document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and total biomass as the focal ecosystem function.
Important terms:
Stage: With seed rain, without seed rainNinitial: Planted species richness
1 Grass1
Clark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.
1.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 2.44 3.60 -4.41 9.57 1.00 2169 2035
## StageWithoutseedrain -12.24 4.04 -20.11 -4.50 1.00 2034 1838
## StageWithseedrain:Shannon 17.72 1.50 14.80 20.59 1.00 2148 2128
## StageWithoutseedrain:Shannon 25.71 1.98 21.89 29.56 1.00 2052 1932
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 23.18 0.60 22.03 24.40 1.00 2921 2567
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2959759 0.02351531 0.2491463 0.34112421.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
1.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
1.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
1.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
1.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 83.58 26.32 32.36 135.89 1.00 5827 2749
## Ninitial4:StageWithseedrain 150.99 24.64 102.15 198.74 1.00 6218 2977
## Ninitial8:StageWithseedrain 167.37 19.75 128.31 206.30 1.00 4548 2418
## Ninitial16:StageWithseedrain 250.82 17.19 216.15 284.27 1.00 5185 2576
## Ninitial32:StageWithseedrain 278.51 20.11 237.99 316.43 1.00 5223 2962
## Ninitial2:StageWithoutseedrain 23.04 10.35 2.65 43.66 1.00 5084 2871
## Ninitial4:StageWithoutseedrain 65.17 15.54 34.07 95.94 1.00 5889 2425
## Ninitial8:StageWithoutseedrain 89.01 14.25 60.70 116.00 1.00 5403 2907
## Ninitial16:StageWithoutseedrain 184.66 17.79 150.14 219.32 1.00 6024 2766
## Ninitial32:StageWithoutseedrain 188.79 23.28 142.48 233.99 1.00 5615 2702
## Ninitial2:StageWithseedrain:Shannon -36.98 16.37 -69.52 -4.99 1.00 5759 2780
## Ninitial4:StageWithseedrain:Shannon -54.03 11.30 -76.05 -32.21 1.00 6206 3023
## Ninitial8:StageWithseedrain:Shannon -47.74 7.43 -62.38 -33.01 1.00 4704 2553
## Ninitial16:StageWithseedrain:Shannon -64.99 5.84 -76.29 -53.37 1.00 5168 2831
## Ninitial32:StageWithseedrain:Shannon -64.73 6.36 -76.81 -51.98 1.00 5074 3036
## Ninitial2:StageWithoutseedrain:Shannon -3.34 6.97 -17.16 10.41 1.00 5037 2791
## Ninitial4:StageWithoutseedrain:Shannon -19.61 7.96 -35.27 -3.88 1.00 5917 2460
## Ninitial8:StageWithoutseedrain:Shannon -23.86 6.33 -35.99 -11.16 1.00 5436 2823
## Ninitial16:StageWithoutseedrain:Shannon -52.74 7.33 -67.11 -38.37 1.00 6061 2596
## Ninitial32:StageWithoutseedrain:Shannon -44.01 8.89 -61.21 -26.23 1.00 5570 2822
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 15.05 0.43 14.25 15.93 1.00 6366 2702
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.699607 0.01080048 0.6768392 0.71936721.2.1 Posterior predictive checks
1.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
1.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
1.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
2 Grass2
Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.
2.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 44.49 1.56 41.45 47.59 1.00 2100 2157
## StageWithoutseedrain 37.55 1.79 33.98 41.03 1.00 2021 2313
## StageWithseedrain:Shannon 8.72 0.56 7.67 9.81 1.00 2048 2289
## StageWithoutseedrain:Shannon 15.06 0.79 13.50 16.60 1.00 2015 2289
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 12.23 0.31 11.64 12.89 1.00 3099 2284
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4411094 0.01965722 0.4009131 0.47816652.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
2.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
2.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
2.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
2.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 85.27 33.54 17.89 150.48 1.00 5967 2907
## Ninitial4:StageWithseedrain 136.35 35.73 65.21 206.05 1.00 5692 2835
## Ninitial8:StageWithseedrain 132.92 32.44 69.06 197.56 1.00 5874 2851
## Ninitial16:StageWithseedrain 92.40 50.17 -4.24 187.40 1.00 4487 2735
## Ninitial32:StageWithseedrain 43.79 61.62 -78.41 164.43 1.00 5436 2580
## Ninitial2:StageWithoutseedrain -16.74 22.64 -60.98 27.64 1.00 5015 2750
## Ninitial4:StageWithoutseedrain 27.61 13.99 0.00 55.05 1.00 5448 2253
## Ninitial8:StageWithoutseedrain 38.52 10.67 17.45 59.66 1.00 5528 2653
## Ninitial16:StageWithoutseedrain 47.39 11.77 23.89 70.57 1.00 4933 2519
## Ninitial32:StageWithoutseedrain 72.57 14.87 42.50 102.05 1.00 5099 2890
## Ninitial2:StageWithseedrain:Shannon -16.30 20.34 -55.82 24.25 1.00 5960 2817
## Ninitial4:StageWithseedrain:Shannon -29.94 15.50 -59.98 1.16 1.00 5683 2813
## Ninitial8:StageWithseedrain:Shannon -20.03 11.20 -42.51 1.89 1.00 5866 2744
## Ninitial16:StageWithseedrain:Shannon -4.68 14.15 -31.68 22.59 1.00 4519 2818
## Ninitial32:StageWithseedrain:Shannon 7.92 14.68 -20.80 36.92 1.00 5437 2672
## Ninitial2:StageWithoutseedrain:Shannon 46.23 13.98 18.61 73.66 1.00 5009 2680
## Ninitial4:StageWithoutseedrain:Shannon 21.86 7.61 7.20 36.94 1.00 5457 2265
## Ninitial8:StageWithoutseedrain:Shannon 17.78 4.81 8.03 27.23 1.00 5501 2690
## Ninitial16:StageWithoutseedrain:Shannon 12.24 4.32 3.76 20.91 1.00 4942 2554
## Ninitial32:StageWithoutseedrain:Shannon 3.26 4.43 -5.63 12.29 1.00 5087 2932
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.05 0.27 8.55 9.61 1.00 6294 3028
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.494272 0.01979012 0.4522059 0.53061772.2.1 Posterior predictive checks
2.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
2.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
2.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
3 Grass3
May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.
3.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 46.13 1.37 43.44 48.76 1.00 1800 2269
## StageWithoutseedrain 46.10 1.49 43.10 49.01 1.00 2113 2319
## StageWithseedrain:Shannon 2.67 0.51 1.70 3.67 1.00 1729 2041
## StageWithoutseedrain:Shannon 2.84 0.59 1.69 4.02 1.00 2190 2300
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.96 0.25 9.49 10.45 1.00 3254 2501
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.06486826 0.01628797 0.03579288 0.098369843.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
3.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
3.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
3.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
3.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 18.84 38.76 -57.94 94.43 1.00 3971 3032
## Ninitial4:StageWithseedrain -7.75 26.33 -59.09 43.28 1.00 3674 2731
## Ninitial8:StageWithseedrain 3.74 26.30 -46.91 55.59 1.00 3700 2938
## Ninitial16:StageWithseedrain 58.09 22.36 13.72 101.67 1.00 4016 2670
## Ninitial32:StageWithseedrain 47.95 31.43 -12.66 108.67 1.00 3796 2821
## Ninitial2:StageWithoutseedrain 64.74 6.51 51.80 77.28 1.00 3622 2826
## Ninitial4:StageWithoutseedrain 25.58 11.03 3.94 46.83 1.00 3971 2806
## Ninitial8:StageWithoutseedrain 36.99 11.79 13.29 60.15 1.00 3568 2554
## Ninitial16:StageWithoutseedrain 64.85 21.20 24.19 106.44 1.00 3837 3038
## Ninitial32:StageWithoutseedrain 79.01 32.15 16.41 142.19 1.00 4088 2839
## Ninitial2:StageWithseedrain:Shannon 18.04 23.23 -27.30 64.11 1.00 3971 3042
## Ninitial4:StageWithseedrain:Shannon 25.42 11.53 3.05 47.68 1.00 3683 2664
## Ninitial8:StageWithseedrain:Shannon 17.48 9.30 -0.66 35.58 1.00 3705 2970
## Ninitial16:StageWithseedrain:Shannon -0.84 6.63 -13.64 12.36 1.00 4030 2792
## Ninitial32:StageWithseedrain:Shannon 2.58 8.26 -13.28 18.71 1.00 3795 2955
## Ninitial2:StageWithoutseedrain:Shannon -9.89 4.04 -17.72 -2.10 1.00 3692 2765
## Ninitial4:StageWithoutseedrain:Shannon 11.56 5.14 1.60 21.52 1.00 3954 2822
## Ninitial8:StageWithoutseedrain:Shannon 6.12 4.49 -2.67 15.14 1.00 3542 2568
## Ninitial16:StageWithoutseedrain:Shannon -3.40 6.67 -16.46 9.46 1.00 3835 3063
## Ninitial32:StageWithoutseedrain:Shannon -5.79 9.13 -23.72 12.00 1.00 4090 2786
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 6.73 0.19 6.38 7.11 1.00 5931 2917
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2337697 0.02421294 0.186099 0.28062913.2.1 Posterior predictive checks
3.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
3.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
3.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
4 Forest1
Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.
4.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 28.07 3.28 21.72 34.60 1.00 2013 2067
## StageWithoutseedrain 30.72 3.81 23.18 38.25 1.00 2158 1845
## StageWithseedrain:Shannon 15.02 1.96 11.09 18.75 1.00 1938 2034
## StageWithoutseedrain:Shannon 13.84 2.80 8.31 19.27 1.00 2093 1872
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 21.88 0.53 20.89 22.93 1.00 2550 2379
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1041526 0.01951819 0.06759033 0.14256554.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
4.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
4.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
4.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
4.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 66.92 11.66 43.47 89.73 1.00 3650 3008
## Ninitial4:StageWithseedrain 89.20 9.86 69.92 108.30 1.00 4907 2476
## Ninitial8:StageWithseedrain 94.26 11.31 71.39 117.04 1.00 4220 2718
## Ninitial16:StageWithseedrain 78.08 14.14 50.81 105.56 1.00 4356 3168
## Ninitial32:StageWithseedrain 41.33 20.33 1.41 80.74 1.00 4013 2704
## Ninitial2:StageWithoutseedrain 65.65 14.48 38.02 94.47 1.00 4459 2669
## Ninitial4:StageWithoutseedrain 75.20 8.97 57.52 93.36 1.00 4337 2669
## Ninitial8:StageWithoutseedrain 100.70 9.79 81.61 119.80 1.00 4512 2932
## Ninitial16:StageWithoutseedrain 58.36 9.52 39.93 77.28 1.00 4300 2634
## Ninitial32:StageWithoutseedrain 46.68 8.74 29.37 63.71 1.00 4358 3062
## Ninitial2:StageWithseedrain:Shannon -24.44 9.77 -43.55 -5.02 1.00 3678 2805
## Ninitial4:StageWithseedrain:Shannon -28.71 7.39 -42.96 -14.02 1.00 4871 2180
## Ninitial8:StageWithseedrain:Shannon -22.95 7.26 -37.50 -8.07 1.00 4152 2894
## Ninitial16:StageWithseedrain:Shannon -6.92 7.11 -20.58 6.81 1.00 4396 3146
## Ninitial32:StageWithseedrain:Shannon 10.07 8.23 -6.00 26.23 1.00 4051 2844
## Ninitial2:StageWithoutseedrain:Shannon -27.94 13.49 -54.69 -2.37 1.00 4467 2770
## Ninitial4:StageWithoutseedrain:Shannon -22.75 7.38 -37.69 -7.93 1.00 4394 2594
## Ninitial8:StageWithoutseedrain:Shannon -33.87 7.33 -48.17 -19.68 1.00 4484 3051
## Ninitial16:StageWithoutseedrain:Shannon 1.76 6.13 -10.56 13.57 1.00 4291 2644
## Ninitial32:StageWithoutseedrain:Shannon 8.29 4.88 -1.26 18.04 1.00 4361 2787
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 17.65 0.50 16.72 18.66 1.01 8326 2825
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.3175321 0.02487158 0.2664479 0.36391344.2.1 Posterior predictive checks
4.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
4.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
4.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
5 Forest2
Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.
5.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 31.86 1.40 29.12 34.62 1.00 2025 1927
## StageWithoutseedrain 28.81 1.72 25.50 32.35 1.00 1814 1787
## StageWithseedrain:Shannon -0.42 0.53 -1.45 0.59 1.00 1954 1721
## StageWithoutseedrain:Shannon -6.13 0.94 -8.05 -4.29 1.00 1785 1837
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 10.62 0.27 10.11 11.19 1.00 3144 2389
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2944755 0.02322893 0.2483561 0.33947655.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
5.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
5.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
5.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
5.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 80.24 10.61 59.37 101.22 1.00 4164 2950
## Ninitial4:StageWithseedrain 101.47 12.73 76.89 127.03 1.00 3905 3046
## Ninitial8:StageWithseedrain 70.65 17.85 35.35 105.04 1.00 3966 3022
## Ninitial16:StageWithseedrain 84.78 30.22 27.69 143.42 1.00 4518 3175
## Ninitial32:StageWithseedrain 116.16 44.15 30.63 201.42 1.00 3862 2871
## Ninitial2:StageWithoutseedrain 14.02 5.80 2.79 25.55 1.00 4454 2848
## Ninitial4:StageWithoutseedrain 3.94 5.44 -6.63 14.58 1.00 4341 3158
## Ninitial8:StageWithoutseedrain 15.49 6.16 3.28 27.85 1.00 3999 2822
## Ninitial16:StageWithoutseedrain 8.31 5.57 -2.23 19.41 1.00 3210 2470
## Ninitial32:StageWithoutseedrain 12.79 7.66 -2.46 27.86 1.00 4103 2956
## Ninitial2:StageWithseedrain:Shannon -31.66 6.98 -45.28 -17.95 1.00 4156 3160
## Ninitial4:StageWithseedrain:Shannon -33.59 6.02 -45.66 -21.88 1.00 3905 2985
## Ninitial8:StageWithseedrain:Shannon -14.83 6.66 -27.50 -1.80 1.00 3983 3030
## Ninitial16:StageWithseedrain:Shannon -16.25 9.03 -33.79 0.87 1.00 4517 3187
## Ninitial32:StageWithseedrain:Shannon -21.66 11.23 -43.46 0.09 1.00 3870 2774
## Ninitial2:StageWithoutseedrain:Shannon 7.47 4.29 -1.08 15.86 1.00 4444 2869
## Ninitial4:StageWithoutseedrain:Shannon 9.14 3.38 2.58 15.72 1.00 4311 3002
## Ninitial8:StageWithoutseedrain:Shannon 0.01 3.09 -6.03 6.13 1.00 4163 2767
## Ninitial16:StageWithoutseedrain:Shannon 2.38 2.61 -2.79 7.41 1.00 3064 2933
## Ninitial32:StageWithoutseedrain:Shannon -0.03 3.08 -6.04 6.06 1.00 4020 2602
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 8.46 0.23 8.00 8.93 1.00 7049 3089
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4971028 0.01972711 0.4556248 0.5333835.2.1 Posterior predictive checks
5.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
5.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
5.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
6 Dryland
Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.
6.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 73.93 1.28 71.42 76.43 1.00 2031 2143
## StageWithoutseedrain 77.93 1.66 74.74 81.21 1.00 2042 2209
## StageWithseedrain:Shannon 1.17 0.54 0.14 2.21 1.00 2048 2236
## StageWithoutseedrain:Shannon -1.54 1.05 -3.66 0.49 1.00 2066 2159
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.38 0.25 8.92 9.87 1.01 3041 2389
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.01443426 0.007817357 0.002762788 0.032485316.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
6.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
6.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
6.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
6.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 63.10 6.65 49.61 76.15 1.00 3401 2635
## Ninitial4:StageWithseedrain 63.05 8.83 46.19 80.79 1.00 3237 2869
## Ninitial8:StageWithseedrain 44.71 11.09 22.75 66.31 1.00 3394 2547
## Ninitial16:StageWithseedrain 42.54 16.03 11.38 74.03 1.00 3825 3004
## Ninitial32:StageWithseedrain 1.27 20.78 -39.27 41.63 1.00 3640 2943
## Ninitial2:StageWithoutseedrain 73.27 4.34 64.79 81.89 1.00 3788 2946
## Ninitial4:StageWithoutseedrain 60.23 4.41 51.65 68.91 1.00 3322 2524
## Ninitial8:StageWithoutseedrain 55.04 4.62 45.93 64.24 1.00 3935 3062
## Ninitial16:StageWithoutseedrain 41.82 6.64 28.58 54.74 1.00 3537 2754
## Ninitial32:StageWithoutseedrain 38.28 9.57 18.90 57.06 1.00 3335 2585
## Ninitial2:StageWithseedrain:Shannon 9.31 4.57 0.34 18.51 1.00 3379 2434
## Ninitial4:StageWithseedrain:Shannon 7.23 4.62 -2.21 15.99 1.00 3219 2914
## Ninitial8:StageWithseedrain:Shannon 13.35 4.58 4.39 22.36 1.00 3376 2708
## Ninitial16:StageWithseedrain:Shannon 11.51 5.42 0.87 22.06 1.00 3828 3014
## Ninitial32:StageWithseedrain:Shannon 21.98 5.99 10.32 33.72 1.00 3637 2925
## Ninitial2:StageWithoutseedrain:Shannon 5.46 3.44 -1.43 12.24 1.00 3819 2942
## Ninitial4:StageWithoutseedrain:Shannon 11.38 3.02 5.34 17.29 1.00 3515 2812
## Ninitial8:StageWithoutseedrain:Shannon 11.96 2.76 6.42 17.32 1.00 3961 3074
## Ninitial16:StageWithoutseedrain:Shannon 16.42 3.56 9.62 23.50 1.00 3537 2676
## Ninitial32:StageWithoutseedrain:Shannon 16.34 4.75 7.01 25.99 1.00 3278 2564
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 7.95 0.23 7.50 8.43 1.00 5706 2674
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.224007 0.02505269 0.1746218 0.27456836.2.1 Posterior predictive checks
6.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
6.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
6.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 